DSpace at VNU: Invariant manifolds of partial functional differential equations

Department of Mathematics, Hanoi University of Science, Khoa Toan, DH Khoa Hoc Tu Nhien,334 Nguyen Trai, Hanoi, Viet Nam b Department of Mathematics and Statistics, York University, Toro

Department of Mathematics, Hanoi University of Science, Khoa Toan, DH Khoa Hoc Tu Nhien,

334 Nguyen Trai, Hanoi, Viet Nam b

Department of Mathematics and Statistics, York University, Toronto, Ont., Canada M3J 1P3

Received April 8, 2003; revised July 22, 2003 Dedicated to the 60th anniversary of the birthday of Professor Toshiki Naito

Abstract

This paper is concerned with the existence, smoothness and attractivity of invariantmanifolds for evolutionary processes on general Banach spaces when the nonlinearperturbation has a small global Lipschitz constant and locally Ck-smooth near the trivialsolution Such a nonlinear perturbation arises in many applications through the usual cut-offprocedure, but the requirement in the existing literature that the nonlinear perturbation isglobally Ck-smooth and has a globally small Lipschitz constant is hardly met in those systemsfor which the phase space does not allow a smooth cut-off function Our general results areillustrated by and applied to partial functional differential equations for which the phase spaceCđơr; 0; Xỡ (where r40 and X being a Banach space) has no smooth inner product structureand for which the validity of variation-of-constants formula is still an interesting open problem

r2003 Elsevier Inc All rights reserved

1 Introduction

Consider a partial functional differential equation in the abstract form

where A is the generator of a C0-semigroup of linear operators on a Banach space X;

F ALđC; Xỡ and gACkđC; Xỡ; k is a positive integer, gđ0ỡ Ử 0; Dgđ0ỡ Ử 0; andjjgđjỡ  gđcỡjjpLjjj  cjj; 8j; cAC :Ử Cđơr; 0; Xỡ and L is a positive number

We will use the standard notations as in [34], some of which will be reviewed inSection 2 As is well known (see[31,34]), if A generates a compact semigroup, thenthe linear equation

center and stable manifolds for Eq (1.1) have been using the so-called LyapunovỜPerron method based on ỔỔvariation-of-constants formulaỖỖ in the phase space C ofMemory[25,26], and as noted in our previous papers (see e.g.[19]), the validity ofthis formula in general is still open The smoothness is an even more difficult issue(even for ordinary functional differential equations) as the phase space involved isinfinite dimensional and does not allow smooth cut-off functions

Much progress has been recently made for both theory and applications ofinvariant manifolds of general semiflows and evolutionary processes (see, forexample, [2Ờ7,10Ờ12,14Ờ17,23,30,32,34]) To our best knowledge, Ck-smoothnesswith kX1 of center manifolds has usually been obtained under the assumption thatthe nonlinear perturbation is globally Lipschitz with a small Lipschitz constantAND is Ck-smooth In many applications, one can use a cut-off function to theoriginal nonlinearity so that the modified nonlinearity satisfies the aboveassumption But if the underlying space does not allow a globally smooth cut-offfunction, as the case for functional differential equations, one cannot get a usefulmodified nonlinearity which meets both conditions: globally Lipschitz with a smallLipschitz constant AND globally Ck-smooth One already faces this problem forordinary functional differential equations, and this motivated the so-called method

of contractions in a scale of Banach spaces by Vanderbauwhed and van Gils [32].This method, together with the variation-of-constants formula in the light of sunsand stars, allowed Dieckmann and van Gils[13]to provide a rigorous proof for the

Ck-smoothness (kX1) of center manifolds for ordinary functional differentialequations

The method of Dieckmann and van Gils[13]has then been extended by Kristin

et al.[22]for the C1-smoothness of the center-stable and center-unstable manifoldsfor maps defined in general Banach spaces The C1-smoothness result was latergeneralized by Faria et al.[16]to the general Ck-smoothness, and this generalizationenables the authors to obtain a center manifold theory for partial functionaldifferential equations Unfortunately, this theory cannot be applied to obtain thelocal invariance of center manifolds as the center manifolds obtained in[16]depend

on the time discretization Moreover, the aforementioned work of Kristin et al.[22]

and Faria et al.[16] is based on a variation-of-constants formula for iterations ofmaps and a natural way to extend these results to partial functional differentialequations would require an analogous formula which, as pointed out above, is notavailable at this stage

We also note that in[6], invariant manifolds and foliations for C1 semigroups inBanach spaces were considered without using the variation-of-constants formula.This work treats directly C1semigroups rather than locally smooth equations, so itsapplications to Eq (1.1) require a global Lipschitz condition on the nonlinearperturbation The proofs of the main results on the C1-smoothness there are based

on a study of the C1-smoothness of solutions to Lyapunov-Perron discrete equations(see[6, Section 2]) Moreover, the main idea in[6, Section 2]is to study the existenceand C1-smooth dependence on parameters of ‘‘coordinates’’ of the unique fixedpoint of a contraction with ‘‘bad’’ characters (in terminology of [6]), that is, thecontraction may not depend on parameters C1-smoothly To overcome this theauthors used the dominated convergence theorem in proving the C1-smoothness ofevery ‘‘coordinate’’ of the fixed point This procedure has no extension to the case of

Ck-smoothness with arbitrary kX1; so the method there does not work for Cksmoothness case As will be shown later in this paper, the Ck-smoothness ofinvariant manifolds can be proved, actually using the well-known assertion thatcontractions with ‘‘good’’ characters (i.e., they depend Ck-smoothly on parameters)have Ck-smooth fixed points (see e.g [21,29]) Furthermore, our approach in thispaper is not limited to autonomous equations, as will be shown later, because itarises from a popular method of studying the asymptoticbehavior of nonautono-mous evolution equations, called ‘‘evolution semigroups’’ (see e.g [8] for asystematicpresentation of this method for investigating exponential dichotomy ofhomogeneous linear evolution equations and[20]for almost periodicity of solutions

-of inhomogeneous linear evolution equations)

An important problem of dynamical systems is to investigate conditions for theexistence of invariant foliations In the finite-dimensional case well-known results inthis direction can be found e.g in [21] Extensions to the infinite-dimensional casewere made in [6,10] In [10] a general situation, namely, evolutionary processesgenerated by a semilinear evolution equations (without delay), was considered.Meanwhile, in[6]a C1-theory of invariant foliations was developed for general C1

semigroups in Banach spaces We will state a simple extension of a result in[6]oninvariant foliations for C1 semigroups to periodicevolutionary processes The

Ck-theory of invariant foliations for general evolutionary processes is still aninteresting question.

In Section 2, we give a proof of the existence and attractivity of center-unstable,center and stable manifolds for general evolutionary processes using the method ofgraph transforms as in[1] Our general results apply to a large class of equationsgenerating evolutionary processes that may not be strongly continuous We then usesome classical results about smoothness of invariant manifolds for maps (described in

[21,28]) and the technique of ỔỔliftingỖỖ to obtain the smoothness of invariant manifolds.The smoothness result requires the nonlinear perturbation to be Ck-smooth,verification of which seems to be relatively simple, in particular, as will be shown

in Section 3, for partial functional differential equations such verification can beobtained by some estimates based on the Gronwall inequality In Section 4 we giveseveral examples to illustrate the applications of the obtained results

We conclude this introduction by listing some notations N; R; C denote the set ofnatural, real, complex numbers, respectively X denotes a given (complex) Banachspace with a fixed normjj jj: For a given positive r; we denote by C :Ử Cđơr; 0; Xỡthe phase space for Eq (1.1) which is the Banach space of all continuous maps from

ơr; 0 into X; equipped with sup-norm jjjjj Ử supyAơr;0jjjđyỡjj for jAC: If acontinuous function x :ơb  r; b ợ d-X is given, then we obtain the mapping

ơ0; dỡ{t/xtAC; where xtđyỡ :Ử xđt ợ yỡ 8yAơr; 0; tAơb; b ợ d: Note that in thenext section, we also use subscript t for a different purpose This should be clear fromthe context

The space of all bounded linear operators from a Banach space X to anotherBanach space Y is denoted by LđX; Yỡ: For a closed operator A acting on a Banachspace X; DđAỡ and RđAỡ denote its domain and range, respectively, and spđAỡ standsfor the point spectrum of A: For a given mapping g from a Banach space X toanother Banach space Y we set

Lipđgỡ :Ử inffLX0 : jjgđxỡ  gđyỡjjpLjjx  yjj; 8x; yAXg:

2 Integral manifolds of evolutionary processes

In this section, we consider the existence of stable, unstable, center-unstable andcenter manifolds for general evolutionary processes, in particular, for semigroups

We should emphasize that the process is not required to have the strong continuity inour discussions below and thus our results can be applied to a wide class ofequations

2.1 Definitions and preliminary results

In this section, we always fix a Banach space X and use the notation Xt to standfor a closed subspace of X parameterized by tAR: Obviously, each Xt is also aBanach space

Definition 2.1 Let fXt; tARg be a family of Banach spaces which are uniformlyisomorphic to each other (i.e there exists a constant a40 so that for each pair t; sARwith 0pt  sp1 there is a linear invertible operator S : Xt-Xs such thatmaxfjjSjj; jjS1jjgoa) A family of (possibly nonlinear) operators Xðt; sÞ : Xs-Xt;ðt; sÞAD :¼ fðt; sÞAR  R : tXsg; is said to be an evolutionary process in X if thefollowing conditions hold:

(i) Xðt; tÞ ¼ It;8tAR; where It is the identity on Xt;

(i) suptARjjPjðtÞjjoN; j ¼ 1; 2; 3;

(ii) P1ðtÞ þ P2ðtÞ þ P3ðtÞ ¼ It;8tAR; PjðtÞPiðtÞ ¼ 0; 8jai;

(iii) PjðtÞUðt; sÞ ¼ Uðt; sÞPjðsÞ; for all tXs; j ¼ 1; 2; 3;

(iv) Uðt; sÞjImP2; Uðt; sÞjImP3ðsÞare homeomorphisms from ImP2ðsÞ and ImP3ðsÞ ontoImP2ðtÞ and ImP3ðtÞ for all tXs; respectively;

(v) The following estimates hold:

jjUðt; sÞP1ðsÞxjjpNebðtsÞjjP1ðsÞxjj; ð8ðt; sÞAD; xAXsÞ;

jjUðs; tÞP2ðtÞxjjpNebðtsÞjjP2ðtÞxjj; ð8ðt; sÞAD; xAXtÞ;

jjUðt; sÞP3ðtÞxjjpNeajtsjjjP3ðsÞxjj; ð8ðt; sÞAD; xAXsÞ:

Note that in the above definition, we define y :¼ Uðs; tÞP2ðtÞx with tXs and xAXtasthe inverse of Uðt; sÞy ¼ P2ðtÞx in P2ðsÞX: The process ðUðt; sÞÞtXsis said to have anexponential dichotomy if the family of projections P3ðtÞ is trivial, i.e., P3ðtÞ ¼0; 8tAR:

Remark 2.3 LetðTðtÞÞtX0be a C0-semigroup of linear operators on a Banach space

X such that there is a t040 for which TðtÞ is compact for all tXt0:As will be shown,this eventual compactness of the semigroup is satisfied by Eq (1.1) with g 0; when

A is the usual ellipticoperator We define a processðUðt; sÞÞtXsby Uðt; sÞ :¼ Tðt  sÞfor allðt; sÞAD: It is easy to see that ðUðt; sÞÞtXsis a linear evolutionary process Wenow claim that the process has an exponential trichotomy with an appropriate choice

of projections In fact, since the operator Tðt0Þ is compact, its spectrum sðTðt0ÞÞconsists of at most countably many points with at most one limit point 0AC: Thisproperty yields that sðTðt0ÞÞ consists of three disjoint compact sets s1;s2;s3;where

s1 is contained infjjzjjo1g; s2 is contained infjzj41g and s3 is on the unit circlefjjzjj ¼ 1g: Obviously, s2 and s3 consist of finitely many points Hence, one canchoose a simple contour g inside the unit discof C which encloses the origin and s1:Next, using the Riesz projection

P1:¼ 12pi

Im Q; where TQðtÞ :¼ QTðtÞQ: Since s2,s3¼ sðTQðt0ÞÞ; TQðtÞ can be extended to agroup on Im Q: As is well known in the theory of ordinary differential equations, in

Im Q there are projections P2; P3 and positive constants K; a; b such that a can bechosen as small as required, for instance aod; and the following estimates hold:

P2þ P3¼ Q; P2P3¼ 0;

jjP2TQðtÞP2jjpKebt; 8t40;

jjP3TQP3jjpKeajtj; 8tAR:

Summing up the above discussions, we conclude that the evolutionary processðUðt; sÞÞtXs defined by Uðt; sÞ ¼ Tðt  sÞ has an exponential trichotomy withprojections Pj; j¼ 1; 2; 3; and positive constants N; a; b0;where

b0:¼ minflog sup

lAs 1jlj; bg;

N¼ maxfK; Mg:

We now give the definition of integral manifolds for evolutionary processes

Definition 2.4 For an evolutionary processðX ðt; sÞÞtXsin X; a set MC,tARfftg 

Xtg is said to be an integral manifold if for every tAR the phase space Xtis split into

M¼ fðt; x; gtðxÞÞAR  X1t  X2tgand

Obviously, if N is an invariant manifold of a semigroupðV ðtÞÞtX0;then R N is anintegral manifold of the evolutionary processðX ðt; sÞÞtXs:¼ ðV ðt  sÞÞtXs:

An integral manifold M (invariant manifold N; respectively) is said to be of class

Ckif the mappings gt (the mapping g; respectively) are of class Ck:In this case, wesay that M (N; respectively) is a integral Ck-manifold (invariant Ck-manifold,respectively)

Definition 2.6 Let ðUðt; sÞÞtXs with Uðt; sÞ : Xs-Xt for ðt; sÞAD be a linearevolutionary process and let e be a positive constant A nonlinear evolutionaryprocess ðX ðt; sÞÞtXs with Xðt; sÞ : Xs-Xt for ðt; sÞAD is said to be e-close toðUðt; sÞÞtXs(with exponent m) if there are positive constants m; Z such that Zemoe and

jjfðt; sÞx  fðt; sÞyjjpZemðtsÞjjx  yjj; 8ðt; sÞAD; x; yAXs; ð2:3Þ

fðt; sÞx :¼ X ðt; sÞx  Uðt; sÞx; 8ðt; sÞAD; xAXs:

In the case where ðUðt; sÞÞtXs and ðX ðt; sÞÞtXs are determined by semigroups ofoperatorsðUðtÞÞtX0andðX ðtÞÞtX0;respectively, we say that the semigroupðX ðtÞÞtX0

is e-close to the semigroup ðUðtÞÞtX0 if the process ðX ðt; sÞÞtXs is e-close toðUðt; sÞÞtXsin the above sense

In the sequel we will need the Implicit Function Theorem for Lipschitz continuousmappings (see[24,28]) which we state in the following lemma

Lemma 2.7 Assume thatX is a Banach space and L : X-X is an invertible boundedlinear operator Let f : X-X be a Lipschitz continuous mapping with

LipðfÞojjL1jj1:Then

(i) ðL þ fÞ is invertible with a Lipschitz continuous inverse, and

Lip½ðL þ fÞ1p 1

jjL1jj1 LipðfÞ;

(ii) if ðL þ fÞ1

¼ L1þ c; thencðxÞ ¼ L1fðL1xþ cðxÞÞ ¼ L1fððL þ fÞ1xÞ; 8xAX

and

jjcðxÞ  cðyÞjjp jjL1jjLipðfÞ

jjL1jj1 LipðfÞjjx  yjj; 8x; yAX: ð2:4Þ

We also need a stable and unstable manifold theorem for a map defined in aBanach space in our ‘‘lifting’’ procedure Let A be a bounded linear operator acting

on a Banach space X and let F be a Lipschitz continuous (nonlinear) operator acting

on X such that Fð0Þ ¼ 0:

Definition 2.8 For a given a positive real r; a bounded linear operator A acting on aBanach space X is said to be r-pseudo-hyperbolic if sðAÞ-fzAC : jzj ¼ rg ¼|: Inparticular, the operator A is said to be hyperbolic if it is 1-pseudo-hyperbolic

For a given r-pseudo-hyperbolicoperator A on a Banach space X we consider theRiesz projection P corresponding to the spectral set sðAÞ-fjzjorg: Let X ¼

Im P"Ker P be the canonical splitting of X with respect to the projection P: Then

we define A1:¼ AjIm P and A2 :¼ AjKer P:

We have

Lemma 2.9 Let A be a r-pseudo-hyperbolic operator acting on X and let F be aLipschitz continuous mapping such that Fð0Þ ¼ 0: Then, under the above notations, thefollowing assertions hold:

(i) Existence of Lipschitz manifolds: For every positive constant d one can find apositive e0; depending onjjA1jj; jjA1

2 jj and d such that ifLipðF  AÞoe; 0oeoe0;then, there exist exactly two Lipshitz continuous mappings g : Im P-Ker P and

h : Ker P-Im P with LipðgÞpd; LipðhÞpd such that their graphs Ws;r:¼grðgÞ; Wu;r:¼ grðhÞ have the following properties:

(a) FWu;r¼ Wu;r;

2 jjjjjA1jjo1 for all 1pjpk; then Wu;ris of class Ck:

Proof For the proof of the lemma, we refer the reader to [27 Section 5; 37,

p 171] &

2.2 The case of exponential dichotomy

This subsection is a preparatory step for proving the existence and smoothness ofinvariant manifolds in a more general case of exponential trichotomy Our latergeneral results will be based on the ones here

2.2.1 Unstable manifolds

We start with the following result:

Theorem 2.10 Let ðUðt; sÞÞtXs be a given linear process which has an exponentialdichotomy Then, there exist positive constants e0;d such that for every given nonlinearprocessðX ðt; sÞÞtXswhich is e-close toðUðt; sÞtXs with 0oeoe0; there exists a uniqueintegral manifold MCR X for the process ðX ðt; sÞtXsdetermined by the graphs of afamily of Lipschitz continuous mappingsðgtÞtAR; gt: X2

t-X1

t with LipðgtÞpd; 8tAR;here X1t;X2t; tAR are determined from the exponential dichotomy of the processðUðt; sÞÞtXs: Moreover, this integral manifold has the following properties:

It is easy to see thatðOd; dÞ is a complete metric space

First of all, we note that using Lemma 2.7 one can easily prove the following:Lemma 2.11 LetðUðt; sÞtXs have an exponential dichotomy with positive constantsN; b and projections P1ðtÞ; P2ðtÞ; tAR as in Definition 2.2 Under the above notations,for every positive constant h0; if

do 12N; eoemh0

then, for every gAOd and ðt; sÞAD such that 0pt  sph0 the mappings

are homeomorphisms

The next lemma allows us to define graph transforms

Lemma 2.12 Let e and d satisfy (2.9) Then, the mapping Gh with 0phoh0 given bythe formula

0oeomin e2mk

2N ;

dðq1 qÞ2ð1 þ dÞ e

Then Gk: Od-Od is a (strict) contraction

The key step leading to the proof of the contractiveness of Gkis the estimate

jjP1ðtÞX ðt; t  kÞx  ðGkgÞtðP2ðtÞX ðt; t  kÞxÞjj

pq0jjP1ðt  kÞx  gtkðP2ðt  kÞxÞjj; 8gAOd; ð2:14Þ

where q0is a constant such that 0oq0o1: Next, for sufficiently small e and d we canapply the above lemmas to prove that the unique fixed point g of Gkin Lemma 2.13

is also a fixed point of Gh provided 0phpk: In fact, for d0ðe; d; hÞ defined by (2.12),there are positive constants e0;d0 such that

(ii) Gx: Od-Od 1;for all 0pxp2k;

(iii) Gk: Od 1-Od 1 and GkðOdÞCOd;

(iv) In Od 1 the operator Gkhas a unique fixed point gAOd:

Thus, for hA½0; k; by the definition of the operator Gkþh (see (2.11)), we have

Ghþk¼ GhGk: Od-Od 1 and Ghþk¼ GkGh: Od-Od 1:Next, for hA½0; k;

Od 1{Ghg¼ GhðGkgÞ ¼ Ghþkg¼ GkðGhgÞAOd 1:

By the uniqueness of the fixed point g of Gkin Od 1;we have Ghg¼ g for all hA½0; k:The above result yields immediately

grðgtÞ ¼ X ðt; sÞðgrðgsÞÞ; 8ðt; sÞAD:

This proves the existence of a unstable manifold M and (i) We now prove (2.5) Let

g¼ ðgtÞtAR be the fixed point of Gk:By (2.14) and the bounded growth

LipðX ðt; sÞÞpKeoðtsÞ; 8ðt; sÞAD;

we can easily show that there are positive constants ˜K and *Z independent of ðt; sÞADand xAX such that

jjP1ðtÞX ðt; sÞðxÞ  gtðP2ðtÞX ðt; sÞðxÞÞjjp ˜Ke*ZðtsÞjjP1ðsÞx  gsðP2ðsÞxjj: ð2:16Þ

To see how (2.5) follows from (2.16), we need the following

Lemma 2.14 Let Y ¼ U"V be a Banach space which is the direct sum of two Banachsubspaces U; V with projections P : Y-U; Q : Y-V; respectively Assume furtherthat g : U-V is a Lipschitz continuous mapping with LipðgÞo1: Then, for any yAY;

dðy; grðgÞÞ :¼ inf

zAU jjy  ðz þ gðzÞÞjjX 1

jjPjj þ jjQjjjjQy  gðPyÞjj: ð2:17ÞProof For any yAY we have

jjyjj ¼ jjPy þ QyjjpjjPyjj þ jjQyjjpðjjPjj þ jjQjjÞjjyjj;

i.e., the normjjyjj
:¼ jjPyjj þ jjQyjj is equivalent to the original norm jjyjj: We havedðy; grðgÞÞ ¼ inf

uAU jjy  ðu þ gðuÞÞjj

Proposition 2.15 Let all the conditions of Theorem 2.10 be satisfied Moreover,assume thatðX ðt; sÞÞtXsis T -periodic(generated by a semiflow, respectively) Then,the family of Lipschitz continuous mappings g¼ ðgtÞtAR has the property that gt¼

gtþT; 8tAR ðgt is independent of tAR; respectively)

Proof Consider the translation St on Od given by ðStgÞt ¼ gtþt;8g; AOd; tAR; tAR: By the T-periodicity of the process ðX ðt; sÞÞtXs (theautonomousness ofðX ðt; sÞÞtXs;respectively) we can show that if g is a fixed point

of Gk;then so is STg (so is Stg; 8tAR; respectively) By the uniqueness of the fixedpoint in Od; we have STg¼ g ðStg¼ g; 8tAR; respectively), completing theproof &

By the above proposition, if ðX ðt; sÞÞtXs is generated by a semiflow, then theunstable integral manifold obtained in Theorem 2.10 is invariant

2.2.2 Stable manifolds

If the processðX ðt; sÞÞtXsis invertible, the existence of a stable integral manifoldcan be easily obtained by considering the unstable manifold of its inverse process.However, in the infinite-dimensional case we frequently encounter non-invertibleevolutionary processes For this reason we will have to deal with stable integralmanifolds directly Our method below is based on a similar approach, developed in

Theorem 2.16 Let ðX ðt; sÞÞtXs be an evolutionary process and let ðUðt; sÞÞtXs be alinear evolutionary process having an exponential dichotomy Then, there exists apositive constant e0 such that if ðX ðt; sÞÞtXs is e-close to ðUðt; sÞÞtXs with 0oeoe0;then, the set

M :¼ fðs; xÞAR  X : lim

is an integral manifold, called the stable integral manifold ofðX ðt; sÞÞtXs; represented

by the graphs of a family of Lipschitz continuous mappings g¼ ðgtÞtAR; where

For a positive constant g let

SðgÞ :¼ fgAS : LipðgÞ :¼ sup

where k is defined by (2.21) and (2.22) Note that½X ðt; t  kÞ1 is, in general, setvalued The next result justifies the use of notations of (2.23) and shows that G is welldefined.

Lemma 2.17 If e040 is sufficiently small, then for every gASðgÞ there is a uniquehASðgÞ such that

Hence, we get the equation for y as follows

y¼ U21½gtðPX ðx þ yÞÞ  QðX ðx þ yÞ  Uðx þ yÞÞ: ð2:24ÞWrite the right-hand side of (2.24) by Fðx þ yÞ; and note that

LipðX ðt; sÞ  Uðt; sÞÞoZemðtsÞ; 8ðt; sÞAD; ð2:25Þwith Zemoe: Then, by definition, for every xAXtk

t ; yAXtk2 ; Fðx; yÞAXtk2 :We nowshow that if Yx:¼ fðu; vÞAXtk1  Xtk2 :jjujjpgjjxjjg; then jjFðx; Þjjpgjjxjj; i.e.,Fðx; Þ leaves Yx invariant In fact,

jjF ðx; yÞjjpy½gjjPXðx þ yÞjj þ pZemkjjx þ yjj;

where

p :¼ supmaxfjjP1ðtÞjj; jjP2ðtÞjjg: ð2:26Þ

Forjjyjjpgjjxjj we have

jjPX ðx þ yÞjjp jjPðXðx þ yÞ  Uðx þ yÞÞjj þ jjPUðx þ yÞjj

Next, we will show that under the above assumptions and notations, Fðx; Þ is acontraction in Yx:In fact, we have

jjF ðx; yÞ  F ðx; y0Þjjp y½jjgtðPX ðx þ yÞÞ  gtðPX ðx þ y0ÞÞjj

Using the assumption on the commutativeness of P with Uðt; sÞ we have

PUðy  y0Þ ¼ P1ðtÞUP1ðt  kÞðy  y0Þ ¼ 0:

We now show that this mapping is Lipschitz continuous with Lipschitz coefficientLipðhtkÞpg: In fact, letting ðx; yÞ and ðx0; y0ÞAX ðt  k; tÞðgrðgtÞÞ; we haveFðx; yÞ  F ðx0; y0Þ ¼ y  y0:Therefore,

jjF ðx; yÞ  F ðx0; y0Þjjp yfjjgtðPX ðx þ yÞÞ  gtðPX ðx0þ y0ÞÞjj

þ pZemkjjðx þ yÞ  ðx0þ y0Þjjg: ð2:31Þ

On the other hand,

jjgtðPX ðx þ yÞÞ  gtðPX ðx0þ y0ÞÞjjp gfPðXðx þ yÞ  Uðx þ yÞÞ

 PðX ðx0þ y0Þ  Uðx0þ y0ÞÞjj

þ jjPUðx þ y  x0 y0Þjjg

p gfyjjx  x0jj þ pZemkg½jjx  x0jj þ jjy  y0jj

¼ gðy þ pZemkÞjjx  x0jj þ gpZemkjjy  y0jj: ð2:32ÞTherefore,

jjy  y0jj ¼ jjF ðx; yÞ  F ðx0þ y0Þjj

p ygpZemkjjy  y0jj þ ypZemkjjy  y0jj

þ ygðy þ pZemkÞjjx  x0jj þ ypZemkjjx  x0jj:

Proof Let g; hASðgÞ and let y :¼ ðGgÞtkðxÞ; y0:¼ ðGhÞtkðxÞ: Then, we have

0Þjj þ pZemkjjy  y0jjg:

On the other hand, we have

jjgtðPX ðx þ yÞ  htðPX ðx þ y0ÞÞjjp jjgtðPX ðx þ yÞ  htðPX ðx þ yÞjj

þ jjhtðPX ðx þ yÞÞ  htðPX ðx þ y0ÞÞjj

p jjPXðx þ yÞjjjjg  hjj

þ gjjPX ðx þ yÞ  PX ðx þ y0Þjj:

We have, usingjjyjjpgjjxjj; that

jjPX ðx þ yÞjjp jjPðXðx þ yÞ  Uðx þ yÞÞjj þ jjPUðx þ yÞjj

jjGg  Ghjj
pyfy þ Zpð1 þ gÞemkg

1 Zypð1 þ gÞemk jjg  hjj
: ð2:35ÞSince 0oyo1; this yields that for small Z40; the graph transform G is a contrac-tion in SðgÞ: &

By the above lemma, for small Z40 the graph transform G has a unique fixed point,say gASðgÞ:

Consider the space B :¼ fv : R-X : suptARjjvðtÞjjoNg and Bj:¼ fvAB : vðtÞ

AIm PðtÞ; 8tARg for j ¼ 1; 2: Let the operators f ; A acting on B be defined by

the formulas

½fvðtÞ :¼ X ðt; t  kÞvðt  kÞ; 8tAR; vAB;

½AvðtÞ :¼ Uðt; t  kÞvðt  kÞ; 8tAR; vAB:

Therefore, for e :¼ Zemk; Ais hyperbolicand Lipðf  AÞpe: We define a mapping

w : B1-B2 by the formula

½wv1ðtÞ :¼ gtðv1ðtÞÞ; 8tAR; v1AB1: ð2:36ÞObviously, LipðwÞpsuptARLipðgtÞ: We want to show that grðwÞ is the stableinvariant manifold of f : We first show that

¼ X ðt; t  kÞðuðt  kÞ; gtkðuðt  kÞÞ; 8tAR:

By Lemma 2.17, since g is the unique fixed point of G; Xðt; t  kÞðuðt  kÞ;

gtkðuðt  kÞÞAgrðgtÞ; i.e., for all AR;

P1ðtÞX ðt; t  kÞðuðt  kÞ; gtkðuðt  kÞÞAIm P1ðtÞand

P2ðtÞX ðt; t  kÞðuðt  kÞ; gtkðuðt  kÞÞ ¼ gtðP1ðtÞX ðt; t  kÞðuðt  kÞ; gtkðuðt  kÞÞ:Hence, if we set

vðtÞ :¼ P1ðtÞX ðt; t  kÞðuðt  kÞ; 8tAR;

then, by definition, vAB1 and fðxÞ ¼ ðv; wðvÞÞAgrðwÞ:

Now we prove

For every yAf1ðgrðwÞÞ; we have f ðyÞAgrðwÞ; and hence, there is uAB1 such that

fðyÞ ¼ ðu; wðuÞÞ: By definition, for every tAR;

Xðt; t  kÞðyðt  kÞÞ ¼ ðuðtÞ; gtðuðtÞÞÞ:

Hence, by Lemma 2.17, yðt  kÞAgrðgtkÞ for all tAR; i.e.,

P2ðt  kÞyðt  kÞ ¼ gtkðP1ðt  kÞyðt  kÞÞ; 8tAR:

Therefore, yAgrðwÞ: Finally, (2.38) and (2.39) prove (2.37)

By Lemma 2.9, for sufficiently small e40; there is a unique Lipschitz mapping

B1-B2 with Lipschitz coefficient less than g whose graph is the unique stableinvariant manifold of the mapping f with Lipðf  AÞoe: By the above discussionand since w : B1-B2is Lipschitz continuous with LipðwÞpg we conclude that grðwÞ

is the stable invariant manifold of f :

Now, forðx; gsðxÞÞAgrðgsÞ; we define

M :¼ fðs; xÞAR  Xjx ¼ AgrðgsÞg

¼ fðs; xÞAR  Xj lim

t-þNXðt; sÞðxÞ ¼ 0g: